lc_circuit

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+ | ====== LC Oscillator Time Domain Analysis ====== | ||

+ | {{:lc_circuit.png?400|}} | ||

+ | |||

+ | ---- | ||

+ | ===== Nodal Equation ===== | ||

+ | |||

+ | \[i_C+i_L=0\] | ||

+ | |||

+ | \[C\frac{dv_{OUT}}{dt}+\frac{1}{L}\int v_{out}=0\] | ||

+ | |||

+ | \[\frac{d^2v_{OUT}}{dt^2}+\frac{1}{LC}v_{OUT}=0\] | ||

+ | |||

+ | ---- | ||

+ | ===== Solution using method of homogeneous and particular solutions ===== | ||

+ | |||

+ | First, notice that the nodal equation is a second order homogeneous equation. | ||

+ | Therefore, we will not need to find a particular solution. | ||

+ | |||

+ | As usual with problems like these, guess that the solution is $e^{st}$. | ||

+ | |||

+ | Plugging in $v_{OUT=}e^{st}$ into the nodal equation yields: | ||

+ | |||

+ | \[e^{st}\left ( s^2 + \frac{1}{LC} \right )\] | ||

+ | |||

+ | Solving the above equation for s gives: | ||

+ | |||

+ | \[s=\pm \sqrt{\frac{-1}{LC}}\] | ||

+ | |||

+ | \[s=\pm j\sqrt{\frac{1}{LC}}\] | ||

+ | |||

+ | So.. | ||

+ | |||

+ | \[v_{out}=e^{j\sqrt{\frac{1}{LC}}t}\] | ||

+ | |||

+ | and | ||

+ | |||

+ | \[v_{out}=e^{-j\sqrt{\frac{1}{LC}}t}\] | ||

+ | |||

+ | There are two properties of homogeneous differential equations that are useful. A constant times the solution is still a solution, and the sum of solutions is still a solution. | ||

+ | |||

+ | Using those two properties we can write: | ||

+ | |||

+ | \[v_{out}=Ae^{j\sqrt{\frac{1}{LC}}t}+Ae^{-j\sqrt{\frac{1}{LC}}t}\] | ||

+ | |||

+ | Any good math student will recognize that is just, | ||

+ | |||

+ | \[v_{out}=A'cos \left [t\sqrt{\frac{1}{LC}} \right ]\] | ||

+ | |||

+ | At $t=0$, $v_{OUT}=v_{IN}$. | ||

+ | |||

+ | Finally we get, | ||

+ | |||

+ | \[v_{out}=v_{IN}cos \left [t\sqrt{\frac{1}{LC}} \right ]\] | ||

+ | |||

+ | ---- | ||

+ | |||

+ | ===== Example ===== | ||

+ | |||

+ | Let's try to make a 1khz oscillator. | ||

+ | |||

+ | Remember that $w=2\pi f$. | ||

+ | |||

+ | \[f=\frac{1}{2 \pi\sqrt{LC}}\] | ||

+ | |||

+ | \[1000=\frac{1}{2 \pi\sqrt{LC}}\] | ||

+ | |||

+ | There are many answers that work...Here is one combination of L and C that produces a 1khz oscillator. | ||

+ | |||

+ | {{:lc_simulation.jpg?800|}} |

lc_circuit.txt ยท Last modified: 2015/06/07 11:06 by admin